PAMoC User's Manual

RIRS | RadRule


[-](rirs|radrule)   rule-number


This keyword (Radial Integration Rule Selection) specifies the rule to be used for numerical integration of the radial function.
The general methodology to calculate the integral of a function f(r) over the interval [0, ∞] spanned by the radial coordinate r, is to map the semi-infinite interval [0, ∞] of the variable r to a finite interval [a, b] of the variable q by a suitable variable transformation r = r(q), and then to use a quadrature formula defined on this interval.
Therefore, each value of the radial-integration-rule selector will combine a specific quadrature rule (qr) with a suitable transformation of the radial coordinate (trc):

radrule = qr + trc

Available quadrature rules and the associated qr-values are reported in Table 1 and are described in section "Methods of Numerical Integration" of this manual.

Table 1. − Available rules for radial quadrature.
quadrature rule[a, b] qr-value (qr + trc)-value
Gauss-Legendre [−1, +1]1015−17
Gauss-Chebyshev of the second kind [−1, +1]20 25−27
Gauss-Gill [0, 1]3031−34, 37
Gauss-Laguerre [0, ∞]4047
Generalized Gauss-Laguerre [0, ∞]5057
Gauss-Hermite [−∞, +∞] 6067
Euler-Maclaurin [a, b]7071−77

The available mappings of the radial coordinate r ∈ [r0, r] into a new coordinate q ∈ [a, b] and the associated trc-values are given in Table 2 and are described in section "Radial integral" of this manual.

Table 2. − Transformations of the radial coordinate.
r ∈ [r0, r]   ⇔   q ∈ [a, b] [a, b] trc-value (qr + trc)-value
MultiExp[0, 1]131, 71
Knowles[0, 1]232, 72
Handy[0, 1]333, 73
Handy (modified)[a, b]4 34, 74
Becke[−1, +1] 515, 25, 75
Ahlrichs[−1, +1]616, 26, 76
linear[a, b]7 17, 27, 37, 47, 57, 67, 77

Because of the restrictions on the interval [a, b], only a few combinations of qr- and trc-values are allowed, and they are reported in the last columns of Tables 1 and 2.

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